Distinct Fringe Subtrees in Random Trees

05/10/2021

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by   Louisa Seelbach Benkner, et al.

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A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in two types of random trees: simply generated trees and families of increasing trees (recursive trees, d-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what `distinct' means) in random trees with n vertices is n/√(log n) for simply generated trees and n/log n for increasing trees.